Hecke.jl is a Julia package for computational algebraic number theory, forming a core component of the OSCAR computer algebra system. It provides a comprehensive suite of algorithms for number fields (absolute, relative, simple, non-simple), orders and ideals, class groups, unit groups, elliptic curves (including isogenies, heights, pairings), quadratic and Hermitian forms, local fields, function fields, associative algebras, and Riemann surfaces. The architecture is modular, with separate source files for each algebraic structure, extensive use of caching via attributes, and integration with GAP and Polymake for group-theoretic and polyhedral computations. The library is designed for research mathematicians and relies on the Nemo/AbstractAlgebra ecosystem for base ring infrastructure. The codebase is large (875 files) and well-tested, with a dedicated test suite covering most modules. It includes reusable components such as sparse matrix types, AVL trees, generic map infrastructure, and factored element representations. The package is actively maintained, with recent commits, CI pipelines, and a changelog documenting ongoing development. While some parts are experimental or contain commented-out code, the core algorithms are production-grade and used in research publications. Hecke.jl is primarily a library (application) rather than a data pipeline, though it includes example scripts and benchmarks that demonstrate research workflows. Its primary purpose is to provide efficient, rigorous implementations of number-theoretic algorithms for use by mathematicians and computer algebra systems.
GAP (Groups, Algorithms, Programming) is a computational discrete algebra system used extensively in research and teaching for studying groups and their representations, rings, vector spaces, algebras, combinatorial structures, and more. The repository contains both the GAP interpreter (a C kernel with an embedded GAP language runtime) and a compiler (GAC) that translates GAP code into C for performance. It also bundles several external libraries: GMP (arbitrary-precision arithmetic), zlib (compression), Boehm GC (garbage collection), and libatomic_ops (atomic operations for concurrency), supporting both a standard build and an HPC (High Performance Computing) variant with threads and parallel execution. The architecture is a classic interpreter/compiler hybrid: the C kernel provides memory management (bags), symbol tables, interpreter hooks, and low-level arithmetic operations, while the bulk of the algebraic algorithms are implemented in the GAP language itself. Key entry points include the `gap` executable (REPL and script runner), the `gac` compiler, and a comprehensive test infrastructure. Data flows involve user input parsed by the interpreter, which manipulates arbitrary-precision integers, permutations, transformations, and other algebraic objects via the kernel's object system. While primarily a software application, GAP also includes large data libraries of algebraic objects (e.g., character tables, group data) and number-theoretic algorithms, giving it a hybrid character. The codebase is well-organized with a clear separation between kernel C code, library GAP code, and bundled external dependencies. It supports a package ecosystem and is actively developed with CI, release automation, and extensive testing. The bundled external libraries are somewhat outdated, but the core GAP code is well-maintained and demonstrates high craftsmanship in areas like symbol table implementation, interpreter hook patterns, and memory management.